What kind of hardware and software are required to build an AI system?
Many different components may go into building an AI system. AI is a broad topic encompassing, machine learning, statistical learning so an AI system could involve anything from a raspberry pi to a spoon. Your title refers specifically to processors of an AI system. The mathematics behind AI can be run on any processor. GPU, CPU, FPGA, and new Neural network chips. A neural network can be run on a Window PC a linux PC AI packages are not OS dependent. I can train a model on my laptop now. 

2019-11-14 20:10:57.503229:
Epoch 1/5
60000/60000 [==============================] - 8s 137us/sample - loss: 0.2925 - acc: 0.9141
Epoch 2/5
60000/60000 [==============================] - 8s 140us/sample - loss: 0.1426 - acc: 0.9574
Epoch 3/5
60000/60000 [==============================] - 9s 142us/sample - loss: 0.1059 - acc: 0.9674
Epoch 4/5
60000/60000 [==============================] - 8s 135us/sample - loss: 0.0862 - acc: 0.9732
Epoch 5/5
60000/60000 [==============================] - 7s 112us/sample - loss: 0.0744 - acc: 0.9763


Just like any computer application an AI needs a processor and memory to store information. Beyond that it is difficult what specifically you are looking for.

A configurable cloud-scale DNN processor for real-time AI